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Discrete Mathematics 306 (2006) 10681071 www.elsevier.com/locate/disc
 

Summary: Discrete Mathematics 306 (2006) 10681071
www.elsevier.com/locate/disc
Explicit construction of linear sized tolerant networks
N. Alona,b,1
, F.R.K. Chungb
aDepartment of Mathematics, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel
bBell Communications Research, 435 South Street, Morristown, New Jersey 07960, USA
Abstract
For every > 0 and every integer m > 0, we construct explicitly graphs with O(m/ ) vertices and maximum degree O(1/ 2),
such that after removing any (1 - ) portion of their vertices or edges, the remaining graph still contains a path of length m. This
settles a problem of Rosenberg, which was motivated by the study of fault tolerant linear arrays.
1988 Published by Elsevier B.V.
1. Introduction
What is the minimum possible number of vertices and edges of a graph G, such that even after removing all but
portion of its vertices or edges, the remaining graph still contains a path of length m? This problem arises naturally
in the study of fault tolerant linear arrays, (see [18]). The vertices of G represent processing elements and its edges
correspond to communication links between these processors. If p, 0 < p < 1 is the failure rate of the processors, it
is desirable that after deleting any p portion of the vertices of G, the remaining part still contains a (simple) path (=
linear array) of length m. Similarly, if , 0 < < 1 denotes the failure rate of the communication links, it is required
that after deleting any portion of the edges of G the remaining part still contains a relatively long path. The objective

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics